When a square fills with people, two numbers usually appear in the news the next morning, and they disagree by a factor of five. Neither side counted anybody.
The method that serious estimates use was worked out in the nineteen sixties by a journalism lecturer in California who could watch demonstrations from his own window. He noticed that the paving of the plaza below was a grid, so he counted the people in a few squares, took an average, and multiplied by the number of squares. Area times density, and nothing else.
The density is where the argument lives. A loose crowd, with room to turn around, runs at roughly one person for every square metre. A dense crowd waiting for a stage is nearer four people to the square metre. Choose the wrong figure and your answer is out by four times before you have measured anything — which is exactly how the same square produces two headlines that cannot both be right.
What the method does give you, reliably, is the order of magnitude. Nobody who has done the arithmetic will tell you a square holding twenty thousand people held a million, because the space for a million does not exist. An estimate that is honest about its own precision is worth far more than a confident number with no working behind it.